Problem: A point $(x,y)$ is randomly picked from inside the rectangle with vertices  $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$.  What is the probability that  $x < y$?
Explanation: The point $(x,y)$ satisfies $x < y$ if and only if it belongs to the shaded triangle bounded by the lines $x=y$, $y=2$, and $x=0$, the area of which is 2.  The rectangle has area 6, so the probability in question is $\dfrac{2}{6} = \boxed{\dfrac{1}{3}}$.

[asy]
defaultpen(.7);
draw((-1,0)--(5,0),Arrow);
draw((0,-1)--(0,3),Arrow);
for (int i=1; i<4; ++i) {
draw((i,-0.1)--(i,0.1));
}
fill((0,0)--(0,2)--(2,2)--cycle,gray(0.7));
draw((-0.1,1)--(0.1,1));
draw((-.1,2)--(0,2));
draw((3,0)--(3,2)--(0,2),linewidth(1.0));
draw((-0.5,-0.5)--(2.8,2.8),dashed);
[/asy]